A unified treatment of some iterative algorithms in signal processing and image reconstruction

Abstract

Let T be a (possibly nonlinear) continuous operator on Hilbert space H. If, for some starting vector x, the orbit sequence {T k x, k = 0, 1, . . .} converges, then the limit z is a fixed point of T ; that is, T z = z. An operator N on a Hilbert space H is nonexpansive (ne) if, for each x and y in H, N x − N y x − y . Even when N has fixed points the orbit sequence {N k x} need not converge; consider the example N = −I , where I denotes the identity operator. However, for any α ∈ (0, 1) the iterative procedure defined by x k+1 = (1 − α)x k + α N x k converges (weakly) to a fixed point of N whenever such points exist. This is the Krasnoselskii-Mann (KM) approach to finding fixed points of ne operators.A wide variety of iterative procedures used in signal processing and image reconstruction and elsewhere are special cases of the KM iterative procedure,for particular choices of the ne operator N. These include the Gerchberg-Papoulis method for bandlimited extrapolation, the SART algorithm of Anderson and Kak, the Landweber and projected Landweber algorithms, simultaneous and sequential methods for solving the convex feasibility problem, the ART and Cimmino methods for solving linear systems of equations, the CQ algorithm for solving the split feasibility problem and Dolidze's procedure for the variational inequality problem for monotone operators.

The viscosity approximation forward-backward splitting method for solving quasi inclusion problems in Banach spaces

“…From [5] we know that F(x) = C T (Cx − d) and F is K-Lipschitz continuous with K the largest eigenvalue of C T C. So we obtain the following result. Theorem 4.2.…”

Strong and weak convergence theorems for split equilibrium problems and fixed point problems in Banach spaces

“…For more results on split feasibility problems, the readers refer to [3,4,6,9,29,34,35]. Now, by Theorem 3.1 and Theorem 4.2, we give the following results on split feasibility problems in Banach spaces: Theorem 5.1.…”

“…Recently, split feasibility problems [3,4,6,9,29,34,35], split variational inequality problems [10,21] and split equilibrium problems [2,17,31] have been investigated by many authors. However, most of the results on these kinds of these problems are investigated only in Hilbert spaces, only a few works are considered in Banach spaces.…”

Iterative algorithms for finding minimum-norm fixed point of a finite family of nonexpansive mappings and applications

“…To solve the MSSFP (4.1), Censor et al [5] first proposed a gradient projection algorithm. However, this iterative algorithm could not reduce to the CQ iterative algorithm [3] for solving the SFP (4.2). Xu [23] proved that the MSSFP (4.1) is equivalent to find a common fixed point of mappings…”